Multivariate discrete distributions via sums and shares

In this article, we develop a sum and share decomposition to model multivariate discrete distributions, and more specifically multivariate count data that can be divided into a number of distinct categories. From a Poisson mixture model for the sum and a multinomial mixture model for the shares, a rich ensemble of properties, examples and relationships arises. As a main example, a seemingly new multivariate model involving a negative binomial sum and Polya shares is considered, previously seen only in the bivariate case, for which we present two contrasting applications. For other choices of the distribution of the sum, natural but novel discrete multivariate Liouville distributions emerge; an important special case of these is that of Schur constant distributions. Analogies and interactions with related continuous distributions are to the fore throughout.